Question

A wave is described by the equation $$y=(1-0mm)\sin \pi (\frac{x}{20cm}-\frac{t}{0-01s})$$ (a) Find the time period and the wavelength . (b) Write the equation for the velocity of the particles. Find the speed of the particle at $x=1\cdot 0cm$ at time $t=0\cdot 01\cdot s$ (c) What are the speeds Of the particles at $x=3\cdot 0cm$, $5.0cm$and $7.0cm$ at $t=0.01s$ ? (d) What are the speeds of the particles at $x=1\cdot 0cm$ at $t=0\cdot 011,0\cdot 012$ and $0.013s$ ?

Answer 1

$y=1mm\sin (\frac{\pi x}{200m}+\frac{t\pi }{1\sec })$

(a) We know, the displacement relation in a wave

$y(x,t)=a\sin (kr-\omega t)$

By comparison we find,

Amplitude $\Rightarrow a=1$mm

Wave number$\Rightarrow k=\frac{\pi }{20}$

Angular frequency$\Rightarrow \omega -\frac{\pi }{1sec}$

Then,

Time period $\Rightarrow T=\frac{2\pi }{\omega }=\frac{2\pi }{\pi /1}=\frac{2\pi }{\pi }\times 1=2s$

Wavelength$\Rightarrow \lambda =\frac{2\pi }{k}=\frac{2\pi }{\pi /20}=\frac{2\pi }{\pi }\times 20=40$cm.

(b) Equation for velocity of particles can be found by differentiating the equation of displacement wrt t

$v=\left(1mm\right)\left(\frac{\pi }{1\sec }\right)\cos (\frac{\pi x}{20cm}+\frac{t\pi }{1sec})$

$x=1.0cmatt=0.01secv=\left(1mm\right)\left(\frac{\pi }{1sec}\right)\cos (\frac{\pi \times 1cm}{20cm}+\frac{\pi \times 0.01\sec }{1\sec })$

$v=\pi mm/s\cos (\frac{\pi }{20}+\frac{\pi }{100})$

$v=\pi mm/s\cos \frac{3\pi }{50}$

$v=\frac{\pi }{1000}\cos \left(\frac{3\pi }{50}\right)m/s$

$\Rightarrow v\approx 3.139\times {10}^{-3}m/s$.

(c) $x=3cm,t=0.01s$

$v=\frac{\pi }{1000}\cos (\frac{3\pi }{20}+\frac{\pi }{100})$

$v=\frac{\pi }{1000}\cos \left(\frac{16\pi }{100}\right)$

$v=\frac{\pi }{1000}\cos \left(\frac{2\pi }{25}\right)$m/s

$v\approx 3.139969822\times {10}^{-3}$ m/s

$x=5cm,t=0.01s$

$v=\frac{\pi }{1000}\cos (\frac{5\pi }{20}+\frac{\pi }{100})$

$v=\frac{\pi }{1000}\cos \left(\frac{26\pi }{100}\right)$

$v=\frac{\pi }{1000}\cos \left(\frac{13\pi }{50}\right)$m/s $\Rightarrow v\approx 3.139681248\times {10}^{-3}m/s$

$x=7cm,t=0.01s$

$v=\frac{\pi }{1000}\cos (\frac{7\pi }{20}+\frac{\pi }{100})$

$v=\frac{\pi }{1000}\cos \left(\frac{36\pi }{100}\right)$

$v=\frac{\pi }{1000}\cos \left(\frac{9\pi }{25}\right)$m/s

$\Rightarrow v\approx 3.139388911\times {10}^{-3}$ m/s

(d) $x=1cm,t=0.011s$

$v=\frac{\pi }{1000}\cos (\frac{\pi }{20}+\frac{11\pi }{1000})$

$v=\frac{\pi }{1000}\cos \left(\frac{61\pi }{1000}\right)m/s$ $\Rightarrow v\approx 3.139982454\times {10}^{-3}$m/s

$x=1cm,t=0.012s$

$v=\frac{\pi }{1000}\cos (\frac{\pi }{20}+\frac{12\pi }{1000})$

$v=\frac{\pi }{1000}\cos \left(\frac{31\pi }{500}\right)m/s$ $\Rightarrow v\approx 3.139981874\times {10}^{-3}$ m/s

$x=1cm,t=0.013s$

$v=\frac{\pi }{1000}\cos (\frac{\pi }{20}+\frac{13\pi }{1000})$

$v=\frac{\pi }{1000}\cos \left(\frac{63\pi }{1000}\right)$m/s $\Rightarrow v\approx 3.139981285\times {10}^{-3}$m/s.